New tightness lower and upper bounds for the standard normal distribution function and related functions

Abstract

Most researches interested in finding the bounds of the cumulative standard normal distribution are not tight for all positive values of the argument . This paper mainly proposes new simple lower and upper bounds for . Over the whole range of the positive argument , the maximum absolute difference between the proposed lower bound and is less than , while it is less than between the proposed upper bound and . Numerical comparisons have been made between the proposed bounds and some of the other existing bounds, which showed that the proposed bounds are more compact than most alternative bounds found in the literature.